Equivariant Splitting: Self-supervised learning from incomplete data

J\'er\'emy Scanvic; Juli\'an Tachella; Patrice Abry; Quentin Barth\'elemy; Victor Sechaud

arxiv: 2510.00929 · v6 · pith:SLILGLWMnew · submitted 2025-10-01 · 💻 cs.CV

Equivariant Splitting: Self-supervised learning from incomplete data

Victor Sechaud , J\'er\'emy Scanvic , Quentin Barth\'elemy , Patrice Abry , Juli\'an Tachella This is my paper

Pith reviewed 2026-05-18 10:50 UTC · model grok-4.3

classification 💻 cs.CV

keywords self-supervised learningequivarianceinverse problemsimage reconstructioninpaintingmagnetic resonance imagingcomputed tomographycompressive sensing

0 comments

The pith

A new definition of equivariance for reconstruction networks makes splitting losses unbiased estimators of the supervised loss from single incomplete observations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that self-supervised training on incomplete data alone can match the performance of supervised methods by combining a splitting loss with a specially designed equivariant reconstruction network. The central result is that this pairing produces an unbiased estimate of the full supervised loss even when each training sample comes from only one incomplete measurement model. This matters for inverse problems where obtaining clean reference data is expensive or impossible. Experiments across image inpainting, accelerated MRI, sparse-view CT, and compressive sensing confirm strong performance when the forward model is highly rank-deficient.

Core claim

The authors introduce a new definition of equivariance tailored to reconstruction networks and prove that, when a network satisfies this property, self-supervised splitting losses yield unbiased estimates of the supervised loss under a single incomplete observation model.

What carries the argument

The new equivariance definition for reconstruction networks, which ensures the splitting loss remains an unbiased estimator of the full supervised loss.

If this is right

Enables training of reconstruction networks without any ground-truth references in highly underdetermined inverse problems.
Delivers state-of-the-art results on image inpainting, accelerated magnetic resonance imaging, sparse-view computed tomography, and compressive sensing.
Applies to settings with a single incomplete observation model where prior self-supervised methods often introduce bias.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach could be tested on inverse problems outside medical imaging, such as astronomical image reconstruction or remote sensing.
Slightly relaxed versions of the equivariance condition might still preserve most of the unbiasedness benefit while easing network design.
Integration with additional physics-based constraints could further improve results on domain-specific tasks.

Load-bearing premise

The new definition of equivariance is sufficient to guarantee that the splitting loss stays unbiased even when only one incomplete observation is available per sample.

What would settle it

A direct numerical check that the self-supervised loss equals the supervised loss for equivariant networks but deviates measurably when the network violates the proposed equivariance condition.

read the original abstract

Self-supervised learning for inverse problems allows to train a reconstruction network from noise and/or incomplete data alone. These methods have the potential of enabling learning-based solutions when obtaining ground-truth references for training is expensive or even impossible. In this paper, we propose a new self-supervised learning strategy devised for the challenging setting where measurements are observed via a single incomplete observation model. We introduce a new definition of equivariance in the context of reconstruction networks, and show that the combination of self-supervised splitting losses and equivariant reconstruction networks results in unbiased estimates of the supervised loss. Through a series of experiments on image inpainting, accelerated magnetic resonance imaging, sparse-view computed tomography, and compressive sensing, we demonstrate that the proposed loss achieves state-of-the-art performance in settings with highly rank-deficient forward models. The code is available at https://github.com/vsechaud/Equivariant-Splitting

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a new equivariance for reconstruction networks that lets splitting losses serve as unbiased estimates of the supervised loss from one incomplete observation.

read the letter

The main thing to know is that this work introduces a tailored equivariance definition for reconstruction networks and shows that pairing it with splitting losses produces unbiased estimates of the full supervised loss even when only a single incomplete measurement is available. That targets a practical setting common in medical imaging and sensing where multiple observations are hard to get. It extends earlier self-supervised inverse-problem methods by focusing on the single-observation case rather than assuming repeated or varied measurements. The experiments cover image inpainting, accelerated MRI, sparse-view CT, and compressive sensing, with reported state-of-the-art performance on these rank-deficient tasks and public code, which makes the claims easier to check. The paper does well at demonstrating practical utility in exactly the regimes where ground truth is expensive or impossible to obtain. The soft spot is the central unbiasedness claim. The abstract states that the new equivariance plus splitting losses yields unbiased estimates, but the derivation steps that would confirm the expectation equality holds for arbitrary rank-deficient operators are not visible in the summary. The stress-test concern about whether the definition forces cross terms to cancel in the null space of A is worth checking directly in the full text; if the proof is clean and the cancellation works without extra assumptions, the result is solid. If it relies on a specific form that does not generalize, the claim weakens. The citation pattern follows standard prior work on self-supervised reconstruction without obvious gaps. This paper is aimed at researchers building networks for incomplete-data inverse problems who want training strategies that avoid ground truth. A reader focused on practical losses for MRI or CT would find the experiments and code release useful. It deserves a serious referee to examine the math and the quantitative comparisons.

Referee Report

2 major / 2 minor

Summary. The paper introduces a new definition of equivariance tailored to reconstruction networks and claims that combining it with self-supervised splitting losses yields unbiased estimates of the supervised loss under a single incomplete observation model. Experiments on image inpainting, accelerated MRI, sparse-view CT, and compressive sensing demonstrate state-of-the-art performance for highly rank-deficient forward models, with code released.

Significance. If the unbiasedness result holds, the work would meaningfully extend self-supervised learning to inverse problems with limited or no ground-truth data, particularly where measurements are severely incomplete. Reproducibility via the linked GitHub repository is a positive factor.

major comments (2)

[§3 and §4] §3 (Method) and §4 (Theory): The proof that the stated equivariance definition implies E[L_split(f_θ, y)] = E[L_sup(f_θ, x)] exactly is not visible in sufficient detail. The derivation must explicitly show how cross terms arising from the rank-deficient measurement operator A cancel under the new equivariance; without this step the central unbiasedness claim remains unverified for arbitrary A.
[Definition 1] Definition 1 (equivariance for reconstruction networks): Clarify whether the definition encodes invariance under the null-space components of A. If it is only of the form f_θ(P y) = P f_θ(y), it is not immediate that this commutes with the splitting operator in a manner guaranteeing unbiasedness for every rank-deficient forward model.

minor comments (2)

[Table 2] Table 2 (quantitative results): Include standard deviations across multiple random seeds or runs to substantiate the SOTA claims.
[Figure 3] Figure 3 (qualitative examples): Add error maps or difference images to make visual comparisons with baselines more precise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive feedback on our manuscript. We address each of the major comments below and have made revisions to improve the clarity of the theoretical contributions.

read point-by-point responses

Referee: [§3 and §4] §3 (Method) and §4 (Theory): The proof that the stated equivariance definition implies E[L_split(f_θ, y)] = E[L_sup(f_θ, x)] exactly is not visible in sufficient detail. The derivation must explicitly show how cross terms arising from the rank-deficient measurement operator A cancel under the new equivariance; without this step the central unbiasedness claim remains unverified for arbitrary A.

Authors: We appreciate the referee highlighting the need for greater explicitness in the derivation. The original Section 4 presented the main steps of the unbiasedness argument, but we agree that the cancellation of cross terms induced by the rank-deficient operator A merits a more granular treatment. In the revised manuscript we have expanded Section 4.2 with a complete, line-by-line derivation: we begin from the definition of the splitting loss, substitute the measurement model y = A x, invoke the new equivariance property to replace f_θ(y) by its projection onto the range of A, and show that all inner-product terms involving the orthogonal complement of range(A) vanish in expectation. The resulting identity E[L_split(f_θ, y)] = E[L_sup(f_θ, x)] therefore holds for any linear A, including those with large null spaces. revision: yes
Referee: [Definition 1] Definition 1 (equivariance for reconstruction networks): Clarify whether the definition encodes invariance under the null-space components of A. If it is only of the form f_θ(P y) = P f_θ(y), it is not immediate that this commutes with the splitting operator in a manner guaranteeing unbiasedness for every rank-deficient forward model.

Authors: We thank the referee for requesting this clarification. Definition 1 is not restricted to the projector form f_θ(P y) = P f_θ(y). It requires that the network output remain unchanged when an arbitrary component from the null space of A is added to its input, i.e., f_θ(y + N z) = f_θ(y) for any z, where N is a basis for null(A). This stronger invariance is precisely what ensures commutation with the splitting operator: because each split measurement is formed by masking in the range of A, the null-space component is invisible to the loss and is therefore automatically preserved by the equivariant network. We have inserted a short remark immediately after Definition 1 that spells out this commutation and its consequence for arbitrary rank-deficient A. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on explicit new definition and mathematical proof

full rationale

The paper introduces a new definition of equivariance tailored to reconstruction networks and derives that self-supervised splitting losses yield unbiased estimates of the supervised loss when this equivariance holds. This follows directly from the stated definition and the expectation over the incomplete observation model without reducing to a fitted parameter, self-referential assumption, or prior self-citation that bears the load. The abstract and claimed results present the unbiasedness as a consequence of the definition rather than by construction or renaming; experiments on inpainting, MRI, CT, and compressive sensing provide external validation. No load-bearing step collapses to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are detailed. The approach appears to rest on standard self-supervised inverse-problem assumptions and the validity of the introduced equivariance definition.

pith-pipeline@v0.9.0 · 5696 in / 1133 out tokens · 32125 ms · 2026-05-18T10:50:38.272493+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce a new definition of equivariance in the context of reconstruction networks, and show that the combination of self-supervised splitting losses and equivariant reconstruction networks results in unbiased estimates of the supervised loss.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1. ... if the matrix Q_A1 ≜ E_g|A1 {(A T_g)^T A T_g} has full rank ... then f^*(y1,A1) = E_{x|y1,A1}{x}.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.